In this thesis, we give an introduction to Arakelov geometry and discuss a possible application to the problem of determining the integral points on an elliptic curve. We first introduce the required notions from algebraic geometry and connections between them. We define the main objects of study in Arakelov geometry, Hermitian vector bundles, and discuss the Arakelov degree, heights, and canonical polygons. As an example, we compute the Arakelov degree and the canonical polygon of the twisting sheaf of Serre endowed with the Fubini-Study metric. Following Bost, Gillet, and Soulé, we also compute the height of projective space. Furthermore, an outlook on a more geometric interpretation of Hermitian vector bundles is given. To build a bridge to elliptic curves, we consider the arithmetic surface attached to a given elliptic curve and define a Hermitian norm on a given vector bundle on this curve. Finally, we take a look at integral points on elliptic curves. After discussing the notion of an integral point on a curve and giving a historical overview of the existing results in this context, we present a modern notion of an integral point on a curve and present an idea of applying Arakelov geometry to the problem of finding an effective bound for the heights of the finitely many integral points on an elliptic curve.